Exercise 1.  Give a parametrization of each contour.  

1 (a).  ,  as indicated in Figure 6.11.

                              Figure 6.11.  The contour  .   Where
                                for  ,    and   
                                for  .

Solution 1 (a).

See text and/or instructor's solution manual.

Answer.     for  ,    and     for  .

Solution.  The contour    consists of  

:  the first quadrant portion of   starting at and ending at ,  and  

:  a line segment starting at    and ending at  .

Since    for    is a parameterization of the circle  ,

we see that    for    will start at  

   and end at  

.  

                    The curve     for   .  

In Section 1.6, we saw that an equation of the straight line segment beginning at and ending at   is:

  

For this exercise we have      and   

                              

And if the parameter is adjusted to be the interval   then we could use

                                for  .  

                    The curve    for  .  

                    Figure 6.11.  The contour  .  Where
                       for  ,  and
                      for  .

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell